Feel free to correct it, but I always imagine it as:"こんにちは、私はキャサリンです。私の日本語が最低です。"

Now, from my understanding, it is not necessary to restate the topic ("私") if it has not changed. Is that true for this situation? And if so, how would the second statement be worded -- exactly the same, but without "私"?

Also, any tips on how to use "です" without being repetitive, but still remaining polite (or at least not rude)?

Because you're introducing yourself, it's presumed that anything you say is part of that introduction, and so unless you specify otherwise, you would remain the topic. If that still left some ambiguity, that you're connecting 'unskilled at japanese' to an apology makes it very clear.

Most of the time, comments out of the blue are going to be taken to be referring to yourself anyway, or else to the person that you're speaking to. Anything said humbly or with an apology or about feeling a certain way is going to be read as being about you ; anything said with honorifics, or as a request, or as a question, is going to be read as being about the person you're speaking directly to. You need to specifically bring up as a topic anybody else, though when there's more than 2 people in a conversation you need to specifically bring up a topic much more.

Anyway, in such a greeting as,

はじめまして。クリスです。日本語がへたですみません、

First it's が, not は as you said above ... the topic of 'me' is implicit, and が is making the japanese language the subject; で is just a continuative form of です、 so that makes 日本語がへた the reason for すみません.

We know the person who is being described as unskilled is the speaker, because,- it's a self introduction- it's a humbling statement (although not 'humble language', there's no special words being used.)- it includes an apology- the previous sentence was a statement of the speaker's identity.

Also, if using that as an introduction, don't forget to add どうぞよろしくおねがいします (if you're the one introducing yourself first) or こちらこそよろしくおねがいします (if you're introducing yourself in reply to an introduction). While they literally mean something like 'please be nice to me' in practice they really mean 'I hope we get on well' and leaving them out suggests that you -don't- hope to get on well with the person.

(There are other variations on the よろしく theme, of course, more humble, more arrogant, less formal, etc.)

Also, with regard to your original phrasing, 最低 isn't generally used like that. It either used literally (the lowest elevation, the lowest percentage) or figuratively to mean 'despicable'. It can hit other connotations of 'lowest/worst' as well, but those are the first ones that jump to mind. (You could, of course, use it to describe your Japanese test score, but I sincerely hope you'll never have to!)

KharismaticKayteh wrote:So with the "Nihongo wa heta de," it is already established that I am the topic, so it is assumed that I am referring to my own Japanese?

Yes. Until you either finish speaking or purposefully introduce a different subject, everything you say will be taken into the context of you introducing yourself. No need to clarify at all, not for Japanese.